A hundred and one years ago, in 1913, the famous British mathematician G. H. Hardy received a letter out of the blue. The Indian (British colonial) stamps and curious handwriting caught his attention, and when he opened it, he was flabbergasted. Its pages were crammed with equations – many of which he had never seen before. There were many kinds of formulas there, and those that first caught his attention had to do with algebraic numbers. Hardy was the leading number theorist in the world – how could he not recognize the identities relating to such numbers, scribbled on the rough paper? Were these new derivations, or were they just nonsensical math scrawls? Later, Hardy would say this about the formulas: “They defeated me completely. I had never seen anything in the least like it before!”

Now, for the first time, mathematicians have identified the mathematics behind these breakthrough scrawls – shedding further light on the genius who made them.

The Mind Behind the Mathematics

The letter was written by Srinivasa Ramanujan, a poverty-stricken, marginally employed young man from the southern-Indian city of Madras, who had little mathematical training but a preternatural ability to derive mathematical identities seemingly out of the void. The equations he sent Hardy came with no proof or theoretical explanation. Ramanujan eventually said that the formulas came to him in a dream, presented as mathematical truths by his family’s goddess, Namagiri Amman (generally known in India as Lakshmi, the wife of Vishnu).

Hardy showed the unusual letter and others that followed to fellow mathematicians, and some told him they believed the writer was a charlatan claiming forgeries as math. But one of them, Percy MacMahon of Cambridge, saw a deep connection between Ramanujan’s work and partitions of numbers he had been working on, and this insight helped convince Hardy that Ramanujan’s writings were both true and new. Studying the letters further, Hardy decided that math identities included in them – involving infinite sums and infinite products, called q-series – were authentic, and that they were very valuable to mathematics as a way of deriving algebraic numbers. It later became clear that the key to Ramanujan’s formulas were two peculiar q-series: the so-called “Rogers-Ramanujan identities,” first studied in the late 1800s by the British mathematician Leonard James Rogers. Hardy concluded about Ramanujan’s identities: “They had to have been written down by a mathematician of the highest class. They must be true because no one would have the imagination to invent them.”

An Abrupt End

Impressed and intrigued by the source of these formulas, Hardy invited Ramanujan to join him in Cambridge. The young Indian accepted the gesture and boarded a ship from Madras to London after receiving permission for the trip from his mother and the family goddess. Hardy and Ramanujan worked together in Cambridge for a few short, intense years, Ramanujan producing hundreds of new mathematical results, which Hardy tried to explain and prove together with him. Hardy would say, late in life, that bringing Ramanujan to England was his greatest achievement as a mathematician.

But Ramanujan was already sick when he arrived in Cambridge, and was frequently hospitalized with various symptoms, initially believed to be of tuberculosis but now concluded to have been a parasitic liver infection. His health deteriorated steadily, and he finally decided to return to India to be close to his family. He died there in 1920, at the young age of 32.

Since then, mathematicians have been fascinated by Ramanujan’s results and there have been many attempts to find the source of his equations that produce algebraic numbers. Freeman Dyson of the Institute for Advanced Study at Princeton reportedly spent the depressing War years in London occupying himself with the study of Ramanujan’s identities. But the origin of the formulas remained a mystery for another 70 years.

Searching for the Mathematical Mother Lode

In April, on the 100th anniversary of Ramanujan’s arrival in Cambridge, the source of his equations was finally found. Ken Ono of Emory University, his graduate student Michael Griffin, and their colleague Ole Warnaar of the University of Queensland presented theorems they had just proved, which vastly generalize the work of Ramanujan and identify the source of his mathematical formulas.

Ono and colleagues discovered that the two Rogers-Ramanujan identities were just specific examples of a literally infinite reservoir of general identities employing similar infinite sums and products. In Ono’s words, they had found the mother lode that gave Ramanujan his “gold nuggets.”

This new, vast ocean of Rogers-Ramanujan-Ono-Griffin-Warnaar identities has the desirable property that it produces algebraic numbers (which are generally hard to obtain) quite readily. One of them is Φ (phi) – the “golden ratio” ubiquitous in art and nature. This number, 1.618…, is the limit of successive terms of the Fibonacci sequence, which even made an appearance in Dan Brown’s book The Da Vinci Code.

Phi was one of the key numbers that occupied Ramanujan’s attention, and the new work paves the way to the discovery of many similar numbers. What the untrained Ramanujan claimed to have gotten from his goddess is seen to constitute one example of a major underlying truth that modern mathematics now possesses – a way of generating similar numbers.

The Mystery Remains

Ramanujan and his work have attracted wide attention. Professor Bruce Berndt of the University of Illinois at Urbana-Champaign has spent 40 years studying Ramanujan’s letters and notebooks – including a “lost notebook” discovered in 1976 – trying to supply proofs for the mathematical results Ramanujan had stated as facts.

“We had a place to start,” he told me, “so our work was somewhat easier – we took Ramanujan’s identities as true, and then proved them.” Then he added: “But the proofs were very hard.” How did Ramanujan know these things were true, how did he come up with such unexpected mathematical facts? “We don’t know Ramanujan’s insights,” he said. “Our proofs are likely much more difficult than the ones he had in his mind.”

So the mystery of how Ramanujan actually obtained his prescient insights about numbers and equations remains even now. The conundrum is reflected in perhaps the most famous story about the Indian number theorist. While Ramanujan was lying in a hospital bed in Putney, England, in 1917, Hardy came to see him. “My taxi had a rather dull number,” Hardy said, just making conversation, “it was 1729.”

“No, Hardy! No, Hardy!” Ramanujan jumped up in bed and exclaimed: “It is a very interesting number! It’s the smallest number expressible as the sum of two cubes in two different ways.” (This is because 1729 = 103 + 93 = 123 + 13.) Ramanujan just naturally knew such things, leaving both proofs and details to others.

Sidebar: What are Algebraic Numbers?

The numbers that come out as solutions of equations such as the ones studied by Ramanujan are of a certain kind, and Φ is one example of them. These are the algebraic numbers. The theory about such numbers is interesting in its own right.

We know many kinds of numbers, and it’s useful to summarize them here. The simplest and earliest-discovered numbers (already known to early humans) are called the natural numbers: 1, 2, 3,…,to infinity, and this set is denoted by N. Then if you add to these numbers zero, you form what mathematicians call a group under addition, which means you can now define additive inverses, i.e., the negative integers. The enlarged set of numbers is called the integers and denoted by Z (from the German for numbers, Zahlen). Add another operation, multiplication, and you now also have multiplicative inverses (except for zero), which are all the fractions, meaning quotients of integers, and the enlarged set is now the field of rationalnumbers, denoted by Q. When you add to this set all the irrational numbers (numbers that can’t be written as quotients of integers), you get the field R of all the real numbers(these are the numbers on the real number line, and we call them “real” to distinguish them from imaginary numbers; if you then also add to them all combinations of the imaginary numbers and real numbers you get the field C of complex numbers).

The German mathematician Georg Cantor proved in the 1800s that while all these sets of numbers are infinite, they are not of the same infinite size. Using ingenious methods, he showed that there are as many rational numbers as there are integers and positive integers. So N, Z, and Q have the same infinite size (or cardinality), while the real numbers, R, have a higher order of infinity (although we don’t know what it is – Cantor’s unprovable conjecture about it is called the continuum hypothesis). The “enlarged infinity” is because of all the irrational numbers – there are just too many of them! We say that N, Z, and Q are countable, while R is uncountable (and so is C).

Algebraic Numbers

But the story gets complicated. Numbers that can be obtained as solutions of equations with rational-number coefficients are called algebraic. So algebraic numbers can be irrational, for example √2. This number is algebraic because it is the solution of the equation x2 – 2 = 0, whose coefficients are all rational (in fact, integer): 1 and -2. Algebraic numbers were one of the main points of interest in Ramanujan’s work. The golden ratio, Φ=1.618…, is irrational but algebraic, because it is the solution of the equation x2 – x – 1 = 0. Ramanujan used infinite sums and products to obtain that number in another way. You can do it by performing the (infinite) operation on your calculator: 1 + 1 = 1/x + 1 = 1/x + 1 = 1/x… and see that the number in the display converges to 1.618… and alternately to 0.618… (which is 1/Φ). This series of operations is the one specified by Ramanujan (although of course he didn’t use a calculator).

One fascinating fact about algebraic numbers is that they are countable, i.e., they have the same order of infinity as N, Z, and Q – even though they are members of the (uncountable) higher set R. Thus their order of infinity is more pedestrian. The hard-core irrational numbers – those that are not algebraic – are called transcendental numbers. These include π (pi) and e. There are “infinitely many more” such numbers than there are algebraic numbers, or integers, or rational numbers!

Squaring the Circle

An interesting fact is that it is because π is transcendental that it’s impossible to square the circle, as the ancients had tried so hard to do. This fact became known only in the nineteenth century, when algebraic numbers became well-understood. It so happens that to square the circle, meaning to construct with straightedge and compass a square whose area is the same as that of a given circle, is tantamount to solving an equation with rational coefficients and getting π as the solution. This is impossible because π is transcendental, and therefore not algebraic. There can never be such an equation that would yield π as a solution.

marie vos savant was a fake, so I dont believe he would want to be associated with a known fraudster.

The tests she took were masterminded(by her boss) such that she could take it until she got them all right. When she knew all the answers to the test, she was tested on the test and got 100%.

Unfortunately this has only recently come to light, to late of course because people now associate savant with smart people, and not frauds.

kat

That’s not what I meant. I meant that this guy was probably a savant.

Jonathan Dunn

nothing on the interwebs. you made that up, didn’t you.

Heimdall222

Apparently made up while smoking a REALLY big blunt…!

daqu

Do you have any evidence for your claim?

Shane Clyburn

The term savant had been used for over 50 years before she was born. They are not related. Nice try though.

tfosorcim

I forget who said that the difference between Richard Feynman and others
whom we characterize as ‘geniuses’, was that Feynman was more than a
genius; he was a magician.

How should we classify Ramanujan, with no formal education?

As Dr Seuss said, “Don’t cry because it’s over. Smile because it happened.”

FreetoSee

His genius was due to the fundamental difference of “knowing” versus “knowledge”, and should speak volumes to those who seek to be Conscious not educated.

EdytaHusseinmuo

my classmate’s aunt makes $68 every hour on the
computer . She has been fired for 7 months but last month her paycheck was
$15495 just working on the computer for a few hours. visit the site R­e­x­1­0­.­C­O­M­

“Mathematics always has hidden surprises, and not just for laymen, but even for mathematicians themselves. This is one of the aspects that make this science wonderful. It is therefore necessary that the peoples of the world strengthen their schools – especially in public education and at the basic level – in the subject of mathematics.”

Sachi Mohanty

A true mathematical genius.

AngeredBovine

If memory serves, Ramanujan’s lost notebook was discovered by George Andrews, not Bruce Berndt.

http://blogs.discovermagazine.com Amir D Aczel

You are right, thank you!! The two worked together, but there is no excuse for my mistake. I’ll ask the editor to correct it a.s.a.p. Thanks!

ChennaiKid

Some time back I was reading a collection of letters relating to Ramanujan, and came across a correspondence between the famous Indian Physics Nobel Laureate Prof. Chandrashekhar and a person whose grandfather knew Ramanujan (both originally from Madras). The person talks about a diary of Ramanujan that went missing after he died, and Chandra confirms it. It couldn’t be the ‘Lost Notebook’ because the letter was dated 1991. In the end both speculate on the possibility of its existence in a dusty attic of some fourth-generation descendant relative of Ramanujan.
Intriguing possibility!

daqu

Happy to see an article about math. But there is almost nothing whatsoever in the article that is remotely new. That Ramanujan was way ahead of his time was already recognized by G.H. Hardy when he received the famous letter from Ramanujan. Bruce Berndt began publishing proofs of Ramanujan’s assertions in 1985. An excellent biography of Ramanujan — “The Man Who Knew Infinity” — came out in 1991. A very well-received play about Hardy and Ramanujan, “A disappearing Number”, premiered in 2007. And on and on.

But the newest Ramanujanlike discoveries, by Ken Ono et al. — which have been published 2006-2013 — are alluded to without being described.

http://blogs.discovermagazine.com Amir D Aczel

Kanigel didn’t write about Ken Ono’s discovery of the infinite set of identities of which Rogers-Ramanujan is one example. The presentation was made in 2014. I think all that is new. I thought that this was a big part of my article.

daqu

“We know many kinds of numbers, and it’s useful to summarize them here.
The simplest and earliest-discovered numbers (already known to early
humans) are called the natural numbers: 1, 2, 3,…,to infinity, and this set is denoted by N. Then if you add to these numbers zero, you form what mathematicians call a group under addition, which means . . ..”

Anyone graduating from college with a bachelor’s in mathematics would instantly know that the set consisting of just the natural numbers along with zero is *not* a group.

http://blogs.discovermagazine.com Amir D Aczel

Read the whole sentence… You form the group when you include the negative numbers. Don’t grab half a sentence and run with it!

daqu

I did read the whole sentence, and the sentence before it, which are verbatim:

“The simplest and earliest-discovered numbers (already known to early humans) are called the natural numbers: 1, 2, 3,…,to infinity, and this set is denoted by N. Then if you add to these numbers zero, you form what mathematicians call a group under addition, which means you can now define additive inverses, i.e., the negative integers”

The last sentence states unequivocally that the natural numbers, when zero is added to them, form a group. Which is not true.

http://blogs.discovermagazine.com Amir D Aczel

You unequivocally can’t read…

daqu

If you see a flaw in my last post, by all means please say what it is, and why.

http://blogs.discovermagazine.com Amir D Aczel

You seem to look for flaws everywhere. Your “post” about there being “nothing new” in this article is especially hurtful and unfair, when–more than any article I’ve done recently–I have gone to a lot of trouble to interview leading mathematicians on very new work (I was only the second journalist to report on this, after one for Scientific American, which was published about the same time). Your point on the definition of a group (another unjustified and highly insulting comment to someone who’s worked with the concept of a group all his professional life) is extremely nitpicky and petty. And you’re obviously no mathematician: If I give you a set, N, and add to it 0 as an additive identity, and give you addition as an operation, and tell you it can be inverted, then all the negative numbers come out of this system automatically. Anyway, this is all I’m going to say to you. If you don’t like my writing (and your hostility is quite evident), surely there is a lot else you can read online or elsewhere.

http://blogs.discovermagazine.com Paul Salevski

Absolutely fascinating. I love math but have no more knowledge about it than the normal layperson. What fascinates me when a “new math” must be created to explain something? How does one go about creating a new math is beyond me when I can hardly explain the basics. Is a new math and new language, a new art form, a new technique… It is hard to describe but it fascinates me nonetheless. And then, when someone just has the mind to truly just understand something while the mathematicians of the world must strive to derive this new math to describe what that person just intuitively knows is astounding.

daqu

From the article: “It so happens that to square the circle, meaning to construct with straightedge and compass a square whose area is the same as that of a given circle, is tantamount to solving an equation with rational coefficients and getting π as the solution.”

No, it is not tantamount to what you say. For example, the cube root of 2 is a root of the polynomial x^3 – 2, yet it is not constructible. Constructibility is subtler than that.

Also, where the article discusses “equations” (as in “Numbers that can be obtained as solutions of equations with rational-number coefficients are called algebraic), it should be instead saying *roots of polynomials with integer (or rational-number) coefficients. Otherwise, there are innumerable cases where the definition of “algebraic number” is false as stated.

http://blogs.discovermagazine.com Amir D Aczel

I am not sure I understand your complaints. I was talking about squaring the circle, and the reason for its impossibility is just as I stated it. Yes, there are other impossible problems, such as doubling the cube (the Delian problem) and trisecting an arbitrary angle. THERE, the reason for the impossibility has to do with Galois theory, and in the case of doubling the cube it boils down, after a lengthy calculation, to the fact that 2 and 3 are mutually prime. As for “integer (or rational-number)” instead of “rational number”–that is also unnecessary. An integer is a rational number: one whose denominator is 1.

http://blogs.discovermagazine.com Amir D Aczel

But, yes, it shouldn’t say “equations” but rather “polynomials” and roots. I was using it more informally, with equation being like the quadratic, with zero on one side.

daqu

You can’t claim “the reason” for a number x not being constructible is that x is transcendental, when plenty of non-transcendental numbers are also not constructible.

Yes, since being algebraic is necessary for being constructible, it is true that if x is transcendental then x is not constructible. But you can’t say that being transcendental is “the reason” — simply because *not* being transcendental does not imply constructibility. As the example of the cube root of 2 shows.

http://blogs.discovermagazine.com Amir D Aczel

I disagree with your logic. Transcendental–>not constructible. There are other reasons for non-constructibility. So what? (Nowhere did I say non-constructible IF AND ONLY IF transcendental). Review your logic for “necessary” and “sufficient”.

daqu

My point is that you wrote “the reason”. The word “the” implies uniqueness. Had you written “a reason,” or better, “it follows,” there would have been no problem with this.

http://blogs.discovermagazine.com Amir D Aczel

“The reason John died was plague; the reason Jane died was cancer.”–Does the “the” here mean that death can only result from Plague (or cancer)? “A reason John died was Plague” means, to me, that he died from several causes.

http://blogs.discovermagazine.com Jane Weir

Your explanation of the relevant mathematical concepts was exceptionally clear and helpful for the non-mathematician.

In your book the Aleph, you mention Decartes claiming dreams significant to his thinking. Ramanujan asserts receiving whispered advice from a goddess. These call to mind a question: How many advances in math and science are claimed to come from such irrational sources by the originators? You are probably one of the few writers who could deal with that subject (in case you have some spare time and don’t know what to do with it).

BTW: According to the website, the movie about Ramanujan is set for release soon.

http://blogs.discovermagazine.com Amir D Aczel

Hi Jane, Yes I find this fascinating…and maybe got attracted to this story because of the knowledge of how Cantor thought (“God told him the continuum hypothesis was true”!). I have no idea how mathematicians obtain results from seemingly-irrational sources. In grad school I once proved a theorem in a dream, but I think it was the conscious mind working overtime while sleeping. Maybe Ramanujan could see these things in a dream but was really proving it in his mind, without knowing. Good research topic for someone in an interdisciplinary math-psychology area. I’ll sure go see the movie when it’s out. Thanks! Ken Ono, apparently, is behind the production, among others.

Michael Weiss

Amir D Aczel : Apropos of nothing, I saw you in person at the Museum of Science Bookclub, right after the discovery of the Higgs boson was announced. What a bravura performance!

I’m afraid daqu has a point, or even two points. The question of squaring the circle could have been expressed more clearly. You write that squaring the circle “is tantamount to solving an equation with rational coefficients and getting π as the solution”; I think anyone who didn’t already know the math would conclude that finding such an equation was sufficient to prove constructibility. Webster’s 7th New Collegiate defines “tantamount” as “equivalent in value, significance, or effect”; etymology is from the French phrase “tant amunter”, “to amount to as much”. To use “tantamount” as you did is poor word choice, to say the least.

As for N, your sentence starts, “Then if you add to these numbers zero, you form what mathematicians call a group under addition”, which clearly says that N+{0} forms a group. You replied, “Read the whole sentence… You form the group when you include the negative numbers. Don’t grab half a sentence and run with it!” But that’s not what the rest of the sentence says! It says that you can define additive inverses, perhaps suggesting that these already belong to N+{0}. Of course you didn’t mean that, but the sentence in no way states that these have to be added to N+{0} to form a group.

I appreciate that phrasing in a blog is sometimes a bit sloppy.I guess it isn’t that easy for you go back and edit the original post.

http://blogs.discovermagazine.com Amir D Aczel

Michael Weiss, thanks!! That was a fun event! I’ll try to improve the writing on that. It was too rushed–after the article itself had been edited many times, the appendix was a quick shot. And I shouldn’t have snapped at responses. I assumed people understood what I am saying without having to be 100% precise. But I should have been.

spirit22g

He received his equations from a much higher dimension, and I’ll bet it all had to do with different layers of consciousness.

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About Amir Aczel

Amir D. Aczel studied mathematics and physics at the University of California at Berkeley, where he was fortunate to meet quantum pioneer Werner Heisenberg. He also holds a Ph.D. in mathematical statistics. Aczel is a Guggenheim Fellow, a Sloan Foundation Fellow, and was a visiting scholar at Harvard in 2005-2007. He is the author of 18 critically acclaimed books on mathematics and science, several of which have been international bestsellers, including Fermat's Last Theorem, which was nominated for a Los Angeles Times Book Award in 1996 and translated into 31 languages. In his latest book, "Why Science Does Not Disprove God," Aczel takes issue with cosmologist Lawrence M. Krauss's theory that the universe emerged out of sheer "nothingness," countering the arguments using results from physics, cosmology, and the abstract mathematics of set theory.